Free power dissipation from functional line integration
Joseph M. Braderaand Matthias Schmidtb,∗aDepartment of Physics, Soft Matter Theory, University of Fribourg, Fribourg, Switzerland;bTheoretische Physik II, Physikalisches Institut, Universit¨at Bayreuth, Bayreuth, Germany
(Received 13 February 2015; accepted 27 March 2015)
Power functional theory provides an exact generalisation of equilibrium density functional theory to non-equilibrium systems undergoing Brownian many-body dynamics. Practical implementation of this variational approach demands knowledge of an excess (over ideal gas) dissipation functional. Using functional line integration (i.e. the operation inverse to functional differentiation), we obtain an exact expression for the excess free power dissipation, which involves the pair interaction potential and the two-body, equal-time density correlator. This provides a basis for the development of approximation schemes.
Keywords: density functional theory; dynamical density functional theory; liquid state structure; Brownian dynamics; perturbation theory
PACS numbers: 61.20.Gy; 64.10.+h; 05.20.Jj
1. Introduction
Classical density functional theory (DFT) provides an exact variational framework for addressing the thermodynam-ics and static microstructure of inhomogeneous fluids in equilibrium [1–3]. Within DFT, the non-trivial many-body physics due to the interparticle interactions follows from the intrinsic excess Helmholtz free-energy functional. The functional is intrinsic in the sense that it is independent of external potential energy fields and hence depends solely on the interparticle interaction potential. However, the ex-cess free energy is only known exactly in a few limiting cases and for special models, such as the low-density func-tional virial expansion and one-dimensional hard-rod model [4]. Approximations are hence unavoidable for any realistic model fluid. Even though the DFT formalism was essen-tially complete by the late 1970s, the development of ap-proximate ‘fundamental measures’ [5] excess free-energy functionals, capable of accurately describing realistic three-dimensional models, required several decades of intense research.
Power functional theory (PFT) has recently been pro-posed as an exact generalisation of DFT for Brownian dynamics of many-body systems out-of-equilibrium [6]. The framework consolidates DFT (which follows when im-posing an equilibrium condition on the power functional) and dynamical density functional theory (DDFT) as special cases into a unified variational scheme. Furthermore, PFT facilitated the development of a non-equilibrium Ornstein–
∗
Corresponding author. Email: Matthias.Schmidt@uni-bayreuth.de
Zernike equation [7,8] for time-dependent two-body corre-lation functions. This has opened up new possibilities for the description of structural relaxation in dense systems and presents intriguing connections with mode-coupling theory. Within PFT, the interparticle interactions are solely con-tained in the intrinsic excess (over ideal gas) free power functional, which is a dynamical analogue of the excess Helmholtz free energy in equilibrium. However, in contrast to the equilibrium case, our knowledge about the intrinsic excess free power is presently very limited. Although some (physically motivated) forms for this functional have been suggested [6], explicit expressions for this central object have so far been lacking.
In this paper, we employ methods of dynamical func-tional calculus in order to gain further insight into the ex-cess intrinsic power functional. In the previous work [6], a physically intuitive splitting of this quantity into the sum of adiabatic and superadiabatic contributions was suggested, where the former is related to the equilibrium free energy and the latter remained unspecified. We here demonstrate how the splitting arises naturally from the application of functional line integration (which is the operation inverse to functional differentiation) and how this leads to a well-defined, statistical mechanical expression for the excess dis-sipation, given in terms of a two-point density correlator. These developments pave the way for systematic approxi-mation schemes, which treat many-body colloidal dynamics at the superadiabatic level.
Published in 0ROHFXODU3K\VLFV ± which should be cited to refer to this work.
The paper is structured as follows. In Section 2, we specify the method of functional line integration, which ap-pears to be an under-used tool for formulating perturbative solutions to variational problems. In Section 3, we show how this method is applied in equilibrium to obtain exact sum rules connecting the free energy with spatial correla-tion funccorrela-tions. In Seccorrela-tion 4, we generalise the methodology to non-equilibrium, working now within the framework of PFT. Our central result is a formally exact expression for the excess dissipation functional (Equations (43) and (44)). This provides a basis for the development of approximation schemes. Finally, in Section 5, we give some conclusions and provide an outlook for future work.
2. Functional line integration
In elementary calculus, integration is the operation inverse to differentiation: a derivative can be integrated up to re-cover the original function to within an integration constant. In the multidimensional case, the gradient of a function can be integrated along a line in the multidimensional space in order to recover, again up to a constant, the original func-tion. (The relationship between a conservative force and its potential energy might serve as a prime physical example.) For singly connected integration domains, the integration line can be chosen arbitrarily.
Here we recall that in functional calculus, there is a sim-ilar inverse operation to functional differentiation, by which a functional can be reconstructed (up to an integration con-stant) from its derivative [9]. As there does not appear to be any standard terminology for this operation, we will hence-forth refer to it as a ‘functional line integral’ (not to be confused with the Feynman– Wiener path integral [10]).
Consider the functional H[g], where g(r) is an arbitrary function of a spatial coordinate r. If we already know the functional derivative
δH [g]
δg(r), (1)
then we can exploit this information to recover the original functional by integration. The operation which achieves this, the functional line integral, is given by
H [g]= H[gr]+ dr g(r) gr(r) d ˜g(r)δH [ ˜g] δ ˜g(r) , (2)
where gr(r) is a reference function. The notation ˜g(r)
de-notes a dummy integration variable and we use the follow-ing short-hand notation:
δH [ ˜g] δ ˜g(r) ≡ δH [g] δg(r) g(r)= ˜g(r) . (3)
The integration in (2) is a standard one-dimensional integral over the value of ˜g at point r, from its reference value, gr(r),
to its target value, g(r). The integral is taken along a path in function space and, provided that H[g] is unique, the result will be path-independent.
Without loss of generality, we choose a linear parametric path through function space:
˜
g(r)≡ gα(r)= gr(r)+ α(g(r) − gr(r)), (4)
where the ‘charging parameter’α varies within the range
0≤ α ≤ 1. Equation (2) thus becomes
H [g]= H [gr]+ 1 0 dα dr g(r)δH [gα] δgα(r) , (5)
where g(r)≡ g(r) − gr(r). The α integration requires
knowledge of the functional derivative for all values ofα,
from the reference (α= 0) to the target (α = 1) function.
The derivative of a vector-valued functional L with vec-torial argument v(r) can be integrated according to
L[v]= L[vr]+ 1 0 dα dr v(r)·δL[vα] δvα(r) , (6)
wherev(r)≡ v(r) − vr(r) is the deviation from the
refer-ence function vr(r) and the integrand is the scalar product
of two vector quantities.
The generalisation of Equation(5) to functionals of functions with multiple arguments is straightforward. For example, the first derivative of the functional G[f(r, r)] can be integrated to obtain G[f ]= G[fr]+ 1 0 dα dr drf (r, r) δG[fα] δfα(r, r) , (7) wheref(r, r)≡ f(r, r)− fr(r, r), and we have again
assumed a linear integration path.
3. Equilibrium
As described very clearly by Evans [2], there are several routes to the equilibrium free energy of an inhomogeneous fluid via integration of appropriate correlation functions. The required mathematical tool for expressing such rela-tions is functional line integration. In this section, we will review and extend (cf. Equation (19)) some equilibrium results, in order to provide context for our subsequent non-equilibrium considerations and to demonstrate how func-tional line integration is a powerful method for constructing approximations.
3.1. Density functional theory
The key object in classical DFT is the grand potential func-tional, given by [1,2] β[ρ ]= Fid [ρ ]+ Fexc [ρ ] − dr (μ− Vext(r))ρ(r), (8)
whereμ and Vext(r) are the chemical and external potentials,
respectively. The first term in Equation (8) is the Helmholtz free energy of the ideal gas:
Fid[ρ]= kBT
dr ρ(r)[ln(3ρ(r, t))− 1], (9)
where kBis Boltzmann’s constant, T is the temperature and
is the thermal wavelength. Fexc[ρ ] is the excess (over
ideal) Helmholtz free energy, which is the only contribution that contains interparticle interactions. The grand potential satisfies the minimum condition
δ[ρ ]
δρ(r) = 0, (10)
at the physically realised equilibrium density distribution. Equation (10) thus generates a formally exact equation for the equilibrium one-body density.
3.2. Integration of the direct correlation
functions
Repeated functional differentiation of the excess Helmholtz free energy with respect to the one-body density generates the hierarchy of direct correlation functions [1,2]. The first member of the hierarchy is the one-body direct correlation function, which is defined by
c(1)(r ; [ρ ])= −δβF
exc
[ρ ]
δρ(r) , (11)
whereβ= (kBT)−1and we make the functional dependence
on the density explicit in the notation. Higher body direct correlation functions are defined by the recursion relation
c(n)(r1, . . . , rn; [ρ ])=
δc(n−1)(r1, . . . , rn−1; [ρ ])
δρ(rn)
. (12)
Functional line integration of the direct correlation functions yields the excess Helmholtz free energy. Choos-ing a linear path through the space of density functions,
ρα(r)= ρr(r)+ α(ρ(r) − ρr(r)), (13)
and integrating Equation (11) yields a formal result for the excess free energy
βFexc [ρ ]= βFexc [ρr]− 1 0 dα drρ(r)c(1) (r; [ρα]), (14) whereρ(r)≡ ρ(r) − ρr(r).
The integrand of Equation (14), which is a functional of the parameterised densityρα(r), can be obtained by
in-tegrating Equation (12), with n= 2, along a linear path,
c(1) (r, [ρα])= c(1)(r ; [ρr]) (15) + α 0 dα drρ(r)c(2) (r, r; [ρα]).
In bulk, the one-body direct correlation function becomes spatially constant and Equation (15) is equivalent to the first of Baxter’s sum rules [11]:
∂c(1)(ρ b) ∂ρb = dr c(2) (r ; ρb), (16)
where the two-body direct correlation function depends only on a single scalar argument.
Substitution of Equation (15) into Equation (14) yields the desired expression for the free-energy functional,
βFexc[ρ ]= βFexc[ρr]− drρ(r)c(1)(r ; [ρr]) − 1 0 dα(1− α) dr × drρ(r)ρ(r)c(2) (r, r; [ρα]). (17)
This is a formally exact result which provides a starting point for approximation schemes. For example, taking the reference density to be that of a bulk system,ρr(r)= ρb=
const, and neglecting the α dependence of the integrand
yields βFRYexc[ρ ]= βF exc [ρb]− drρ(r)c(1)(ρb) −1 2 dr drρ(r)ρ(r)c(2)(|r−r|; ρ b), (18) which is the well-known quadratic functional, first em-ployed by Ramakrishnan and Youssouff (RY) in their study of crystallisation [12]. Equation (18) requires as input the bulk two-body direct correlation function, available for ex-ample, from integral equation theory [3].
Equation (18) is identical to a truncated functional Tay-lor expansion of the excess free energy. The remainder,
given by the sum of all terms of cubic and higher order, is known as the ‘bridge functional’ [13]. A closed-form ex-pression for this quantity is found by integration of Equa-tion (12), with n = 3, and substitution of the result into Equation (17). This yields
βFexc [ρ]=βFexc RY[ρ ]− 1 2 1 0 dα(1−α)2 dr dr dr × ρ(r)ρ(r)ρ(r)c(3) (r, r, r; [ρα]), (19)
where the second term is the bridge functional.
3.3. Integration of the pair density
A further exact expression for the excess free energy can be obtained by integrating the two-body density correlator [2]. Starting from the grand canonical partition function, it is straightforward to show that
δFexc [ρ ] δφ(r, r) = 1 2ρ (2) (r, r), (20)
where φ(r, r) is the interaction pair potential and
ρ(2)(r, r) is the two-body density [3]. (Note that Equation
(20) remains valid if triplet and higher body interparticle interactions are present.) Functional line integration in the space of pair potentials yields
βFexc [ρ ]= βFexc [ρr] +1 2 1 0 dα dr drφ(r, r)ρ(2) (r, r; [φα]), (21)
where the choiceφα(r, r)= φr(r, r) + αφ(r, r)
en-ables a part of the interaction potential to be ‘turned on’ by increasingα. Here φ(r, r)= φ(r, r)− φr(r, r) is
the difference between the full pair potential and that of the reference system.
Equation (21) forms the basis of virtually all liquid state perturbation theories. Neglecting the dependence of the in-tegrand onα yields a first-order theory [14]. Furthermore, if the two-body density of the reference system is factorised,
ρ(2)(r, r)= ρ(r)ρ(r)g(2)(r, r)≈ ρ(r)ρ(r), such that the
in-homogeneous pair correlation function g(2)(r, r)≈ 1, then
we recover the familiar mean-field approximation [1,2]
βFexc[ρ]= βFexc[ρr]+ 1 2 dr drφ(r, r)ρ(r)ρ(r), (22) which is frequently employed to treat the attractive part of the interparticle attraction,φ(r, r)≡ φatt(r, r), as a
perturbation to the hard-sphere reference system.
4. Non-equilibrium
The PFT [6] is the generalisation of DFT to treat non-equilibrium systems undergoing Brownian dynamics. Derivatives of the power functional with respect to external, density or current fields are related to dynamic correlation functions [7,8]. We will here focus on functional line inte-gration with respect to the current.
4.1. Power functional theory
The central mathematical object within PFT is the free power functionalRt[ρ, J], which depends on time t and is
a functional of the density and current [6]. One can split off the external contributions,
Rt[ρ, J]= Wt[ρ, J]− dr (X(r, t)−∇Vext(r, t))· J(r, t) + dr ˙Vext(r, t) ρ(r, t), (23)
where X(r, t) is a non-conservative external force and the overdot indicates a time derivative of the external potential, which can be time-dependent. The intrinsic part Wt[ρ, J] of
the power functional can be split into two contributions:
Wt[ρ, J]= Wtid[ρ, J]+ Wtexc[ρ, J], (24)
where the non-interacting, ideal intrinsic power is given by the sum of dissipative and adiabatic contributions:
Wid t [ρ, J]= drγ J 2 (r, t) 2ρ(r, t) + dr J(r, t)· ∇δF id [ρ ] δρ(r, t), (25) whereγ is the friction coefficient (related to the bare
dif-fusion coefficient byγ = kBT/D0), and the ideal gas
free-energy functional is given by Equation (9). The excess part of the intrinsic power depends solely upon the interparticle interactions and thus specifies the system under investiga-tion.
The variational principle of PFT implies the minimisa-tion
δRt[ρ, J]
δJ(r, t) = 0, (26)
where the functional derivative is taken at fixed time, and
ρ(r, t) and J(r, t) are treated as independent fields. In
equi-librium, Equation (26) reduces to Equation (8), thus recov-ering DFT as a special case. Application of Equations (26)– (23) yields a force balance equation as follows:
γ J(r, t) ρ(r, t) = −kBT∇ ln 3ρ(r, t)+ X(r, t) − ∇Vext(r, t) −δWtexc[ρ, J] δJ(r, t) . (27)
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When supplemented by the continuity equation,
∂ρ(r, t)
∂t = −∇ · J(r, t), (28)
Equation (27) provides a formally exact equation of mo-tion for the one-body density. However, the excess intrin-sic power functional, containing the non-trivial phyintrin-sics, re-mains to be determined. Expressing this quantity in terms of statistical mechanical correlation functions is the objective of the following section.
4.2. Functional line integration in PFT
Many-body Brownian dynamics can be described using the Smoluchowski equation of motion for the N-body con-figurational probability distribution [15]. Integrating the Smoluchowski equation over all but one of the particle coordinates yields an exact equation for the current [16]:
γ J(r, t) ρ(r, t) = −kBT ∇ ln 3ρ(r, t)+ X(r, t) − ∇V ext(r, t) − drρ (2)(r, r, t) ρ(r, t) ∇φ(r, r ), (29)
whereρ(2)(r, r, t) is the non-equilibrium equal-time density
correlation function. Comparison of Equation (27) with Equation (29) enables the identification
δWexc t [ρ, J] δJ(r, t) = drρ (2)(r, r, t) ρ(r, t) ∇φ(r, r ), (30)
which makes explicit the fact that the functional derivative of the excess intrinsic power with respect to the current generates (up to a minus sign) the interparticle interaction forces acting at position r. The right-hand side of Equation (30) is easily generalised to account for triplet and higher body potential interactions [16], but for clarity of presenta-tion, we restrict ourselves to systems with pairwise additive interaction potentials.
Taking Equation (30) as a starting point, we perform a functional line integral to reconstruct the excess intrinsic free power functional. We choose the reference system to be an equilibrium state, Jr(r, t)= 0, and integrate along the
parameterised path Jα(r, t)= αJ(r, t), from α = 0 to α =
1, while holding the density fixed. This yields
Wexc t [ρ, J]= 1 0 dα dr J(r, t)·δWt[ρ, Jα] δJα(r, t) , (31)
where we have used the fact that Wexc
t [ρ, 0 ]= 0 (i.e. for
an equilibrium system). The integration in Equation (31) is only over spatial coordinates, because the derivative is taken at fixed time. Substitution of Equation (30) into
Equation (31) leads to Wtexc[ρ, J]= 1 0 dα dr drρ (2)(r, r, t ; [ρ, J α]) ρ(r, t) × J(r, t) · ∇φ(r, r), (32)
where we have indicated the functional dependence in the notation explicitly. Equation (32) constitutes an exact re-lation of the excess intrinsic power functional to a well-defined equal-time pair correlator.
The integral expression (32) has the appealing property that the well-established DDFT is recovered by neglecting the dependence of the integrand on the charging parameter
ρ(2)
(r, r, t ; [ρ, Jα])≈ ρ(2)(r, r, t ; [ρ, 0 ]). (33)
The right-hand side of Equation (33) defines the adiabatic two-body density,ρad(2)(r, r, t , [ρ ])≡ ρ
(2)(r, r, t ; [ρ, 0 ]),
namely the two-body density of an equilibrium system with one-body density profileρ(r, t). Substituting Equation (33)
into the force integral (30) enables us to use the following equilibrium sum rule [1]:
drρ (2) ad(r, r, t; [ρ ]) ρ(r, t) ∇φ(r, r )= ∇δFexc[ρ ] δρ(r, t) , (34)
which relates the interaction forces to the excess Helmholtz free energy. We thus obtain the DDFT approximation to the excess intrinsic power functional
Wtexc[ρ, J] ≈
dr J(r, t)· ∇δF
exc[ρ ]
δρ(r, t) . (35)
Equations (27), (28) and (35) recover the familiar DDFT equation for the one-body density [16,17].
4.3. Exact excess dissipation functional
We now proceed to further develop the exact expression (32). We will see that this leads to a splitting of the excess intrinsic power into the sum of adiabatic and superadiabatic contributions. Defining the interaction force,
f(r, t ; [ρ, J])≡ −δW
exc t [ρ, J]
δJ(r, t) , (36)
enables Equation (31) to be rewritten as
Wexc t [ρ, J]= − 1 0 dα dr J(r, t)· f(r, t ; [ρ, Jα]). (37)
The parameterised force, f(r, t ; [ρ, Jα]), that appears
in the integrand of Equation (37) can be found by integra-tion of its derivative (starting from an equilibrium reference
state). At this point, it is important to clarify the nature of the functional derivative. The expression (31) involves a space integral, but no time integral, in recognition of the fact that Equation (26), and thus Equation (30), is a func-tional derivative taken at fixed time. However, when using functional line integration to formally re-express f(r, t ; [ρ, Jα]), we are no longer constrained by the fixed-time
differentiation of the power functional variational princi-ple. The most general choice is to consider functional line integration of the full functional derivative. This yields
f(r, t ; [ρ, Jα])= f(r, t ; [ρ, 0 ]) + α 0 dα dr t t0 dtJ (r, t)·δf(r, t[ρ, Jα]) δJα(r, t) , (38) where t0is an initial time.
Substitution of Equation (38) into Equation (37) then enables the exact expression (32) for the excess intrinsic power functional to be re-expressed as a sum of two terms:
Wexc t [ρ, J]= − dr J(r, t)· f(r, t ; [ρ, 0 ]). − 1 0 dα(1− α) dr dr t t0 dtJ(r, t)J(r, t) : ×δf(r, t ; [ρ, Jα]) δJα(r, t) , (39)
where the colon represents a full contraction between the current–current dyadic and the second-rank tensor func-tional derivative. The first term in Equation (39) involves the interaction force arising from an inhomogeneous den-sity distribution in equilibrium (vanishing current). Us-ing Equations (30), (34) and (36)), we obtain the exact expression Wexc t [ρ, J]= dr J(r, t)· ∇δF exc[ρ ] δρ(r, t) − 1 0 dα(1− α) dr dr t t0 dtJ(r, t)J(r, t) : ×δf(r, t ; [ρ, Jα]) δJα(r, t) , (40)
where the first term now explicitly accounts for the adia-batic intrinsic power. By employing the continuity equation and the functional chain rule, this adiabatic contribution can be identified as the time derivative of the excess free-energy functional, ˙Fexc[ρ ]. The second, ‘superadiabatic’
term describes the excess dissipation due to interparti-cle interactions. In deriving Equation (40), we have thus found a natural splitting of the excess intrinsic power into
adiabatic and superadiabatic contributions:
Wexc
t [ρ, J]= ˙Fexc[ρ ]+ Ptexc[ρ, J], (41)
where the excess dissipation functional,Pexc
t [ρ, J],
corre-sponds to the second term in Equation (40). The decompo-sition (41) is identical to the physically motivated splitting introduced in [6].
Differentiation of Equation (36) with respect to the cur-rent, at fixed time and using Equation (30) enables the func-tional derivative of the force appearing in Equation (40) to be expressed in terms of the two-body density correlator:
δf(r, t; [ρ, J]) δJ(r, t) = dr δρ (2)(r, r, t ; [ρ, J]) δJ(r, t) ∇φ(r, r) ρ(r, t) , (42) where the integrand is a dyadic product of two vector functions. Substitution of Equation (42) into Equation (40) yields our final form for the exact excess dissipation func-tional, Pexc t [ρ, J]= dr dr t t0 dtJ(r, t)· × K(r, r, t, t; [ρ, J])· J(r, t), (43)
where the dyadic matrix Kernel is given exactly by
K(r, r, t, t; [ρ, J])= × 1 0 dα(1−α) dr δρ (2)(r, r, t ; [ρ, J α]) δJα(r, t) ∇φ(r, r) ρ(r, t) . (44) Equations (43) and (44) constitute the main result of the present paper. The excess dissipation functional accounts for the fact that the pair correlations are not equilibrated during the time evolution of the system, and hence corrects the adiabatic approximation. The kernel (44) depends ex-plicitly on the interaction forces and thus, when substituted into Equation (43), corrects the ideal gas dissipation given by the first term in Equation (25). It is intuitive that scat-tering events arising from the interaction forces will lead to a coupling between different components of the current vector, and this is reflected in the tensorial character of the kernel. For systems under external driving, the ability of the excess dissipation functional to capture non-affine, interaction-induced flows will, in general, be essential to ob-tain physically realistic predictions (see e.g. [18,19], which highlight some situations where DDFT fails).
Substitution of the split excess intrinsic power (41) into the force balance (27) shows that the interaction-induced dissipative force (‘superadiabatic’ force) is given by the
derivative
Fsad(r, t)= −δP
exc
t [ρ, J]
δJ(r, t) , (45)
with the excess dissipation given by Equation (43). The superadiabatic force has been studied in detail for a simple one-dimensional model, where a non-trivial dependence on particle density was revealed [20].
4.4. Perturbation approximation
By neglecting theα-dependence of the integrand (44) and
choosing a zero reference current, we obtain the following approximation to the kernel:
K0(r, r, t, t; [ρ]) = drδρ (2)(r, r, t ; [ρ, J]) δJ(r, t) J=0 ∇φ(r, r) 2ρ(r, t) , (46)
which is a functional of the instantaneous density, but inde-pendent of the current. We have thus constructed a leading-order (in current) perturbation theory about the adiabatic state. Equation (46) requires as input the flow-distorted pair correlations close to equilibrium.
It is interesting to note the similarity of the pertur-bative excess dissipation functional, obtained by substi-tuting Equation (46) into Equation (43), with the exact non-interacting form (the first term in Equation (25)). This suggests to revert to the total dissipation as the sum of ideal and excess contributions P [ρ, J]= Pid
t [ρ, J]+
Pexc
t [ρ, J], such that the perturbation approximation to this
quantity takes the following form:
P0 t[ρ, J]= dr dr t t0 dt × J (r, t)J(r, t)2ρ(r, t) : γ 0 (r, r, t, t; [ρ ]), (47)
where the effective friction tensor depends solely on the density and is given by
γ0(r, r, t, t; [ρ ])= γ δ(r − r)δ(t− t)1 + drδρ (2)(r, r, t ; [ρ, J]) δJ(r, t) J=0 ∇φ(r, r). (48)
The first term in Equation (48) generates the exact non-interacting dissipation, while the second term incorpo-rates the spatial non-locality arising from interparticle interactions.
5. Conclusions
In the framework of PFT, we have used the method of func-tional line integration to derive a formally exact expres-sion for the excess power dissipation functional, given by Equations (43) and (44). The excess dissipation requires as input information about how the inhomogeneous two-body density is distorted by flow (i.e. a non-vanishing current). It is precisely this feature of the non-equilibrium microstruc-ture which is not resolved in the standard, adiabatic DDFT approximation. By neglecting the ‘charging’ integral in Equation (44), we have generated a perturbation theory about the adiabatic state, characterised by the kernel (46). Given that the adiabatic approximation often performs quite well, we anticipate that using a perturbative excess dissipa-tion funcdissipa-tional will generate the most important correcdissipa-tions to DDFT.
In Section 4.3, we employed functional line integra-tion to decompose the excess intrinsic power into a sum of adiabatic and excess dissipation terms. This is certainly a convenient choice for the application of the theory, because it enables the many excellent approximations for the equi-librium free energy to be exploited, but does not represent a fundamental constraint on the theory. The power func-tional approach is a general and exact dynamical theory, which does not have to make any reference to the adiabatic state. This independence of any adiabatic assumption is made explicit in Section 4.2, where we formulate an exact expression, Equation (32), for the excess intrinsic power,
Wt[ρ, J]. This quantity could be made the focus of direct
approximation schemes and is the most fundamental ob-ject to be approximated within PFT. It is only when one chooses to formulate the theory about the adiabatic state, thus adopting DDFT as a zeroth-order approximation, that the excess dissipated power,Pexc
t [ρ, J], becomes the main
object of attention.
For bulk systems, it is known that the flow-induced distortion of the pair correlations contributes to both the shear and bulk viscosities [21]. For general inhomogeneous situations, the excess power dissipation thus takes account of these interaction-induced contributions, which are absent from the ideal dissipation functional and are consequently absent from DDFT. Our formal results provide a solid basis for the development of practical approximation schemes. The next step in our research programme will be to develop and test explicit approximations to the excess dissipation for simple model systems. Work along these lines is in progress.
Acknowledgements
It is a great pleasure to dedicate this paper to Jean-Pierre Hansen, whose work has shaped our thinking about physics in many useful ways.
Disclosure statement
No potential conflict of interest was reported by the authors.
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